What are the divisors of 528?

1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 33, 44, 48, 66, 88, 132, 176, 264, 528

There is a total of 20 positive divisors.
The sum of these divisors is 1488.
The arithmetic mean is 74.4.

16 even divisors

2, 4, 6, 8, 12, 16, 22, 24, 44, 48, 66, 88, 132, 176, 264, 528

4 odd divisors

1, 3, 11, 33

How to compute the divisors of 528?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 528 by each of the numbers from 1 to 528 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

528 / 1 = 528 (the remainder is 0, so 1 is a divisor of 528)
528 / 2 = 264 (the remainder is 0, so 2 is a divisor of 528)
528 / 3 = 176 (the remainder is 0, so 3 is a divisor of 528)
...
528 / 527 = 1.0018975332068 (the remainder is 1, so 527 is not a divisor of 528)
528 / 528 = 1 (the remainder is 0, so 528 is a divisor of 528)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 528 (i.e. 22.978250586152). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

528 / 1 = 528 (the remainder is 0, so 1 and 528 are divisors of 528)
528 / 2 = 264 (the remainder is 0, so 2 and 264 are divisors of 528)
528 / 3 = 176 (the remainder is 0, so 3 and 176 are divisors of 528)
...
528 / 21 = 25.142857142857 (the remainder is 3, so 21 is not a divisor of 528)
528 / 22 = 24 (the remainder is 0, so 22 and 24 are divisors of 528)